Sixty-Four Curves of Degree Six

Transcription

1 Sixty-Four Curves of Degree Six Bernd Sturmfels MPI Leipzig and UC Berkeley with Nidhi Kaihnsa, Mario Kummer, Daniel Plaumann and Mahsa Sayyary 1 / 24

2 Poset 2 / 24

3 Hilbert s 16th Problem Classify all real algebraic curves of degree d in the plane P 2 R. Stratify the parameter space P d(d+3)/2 R. Assume that the complex curve in P 2 C is smooth. Complete answers are known up to d = 7, due to Harnack, Hilbert, Rohn, Petrovsky, Rokhlin, Gudkov, Nikulin, Kharlamov, Viro,... Two curves C 1 and C 2 have same topological type if some homeomorphism of P 2 R P2 R restricts to a homeo C 1 C 2. 3 / 24

4 Hilbert s 16th Problem Classify all real algebraic curves of degree d in the plane P 2 R. Stratify the parameter space P d(d+3)/2 R. Assume that the complex curve in P 2 C is smooth. Complete answers are known up to d = 7, due to Harnack, Hilbert, Rohn, Petrovsky, Rokhlin, Gudkov, Nikulin, Kharlamov, Viro,... Two curves C 1 and C 2 have same topological type if some homeomorphism of P 2 R P2 R restricts to a homeo C 1 C 2. Finer notion of equivalence comes from the discriminant, a hypersurface of degree 3(d 1) 2 in P d(d+3)/2 R. Points on are singular curves. The rigid isotopy types are the connected components of P d(d+3)/2 R \. Two curves C 1 and C 2 in the same rigid isotopy class have the same topological type... the converse is not true for d 5. 4 / 24

5 Sextics Theorem (Rokhlin-Nikulin Classification) Our paper: d = 6 The discriminant of plane sextics is a hypersurface of degree 75 in P 27 R. Its complement has 64 connected components. The 64 rigid isotopy types are grouped into 56 topological types, with number of ovals ranging from 0 to 11. The distribution equals # ovals all Rigid isotopy Topological The 56 types are seen in our poset. Rokhlin (1978) carried out the classification of rigid isotopy types. Nikulin (1980) completed the proof in his study of real K3 surfaces. 5 / 24

6 14 Are Dividing The following eight types consist of two rigid isotopy classes: (41) (21)2 (51) 1 (31)3 (11)5 (81) (41)4 9. The six maximal types necessarily divide their Riemann surface: (91)1 (51)5 (11)9 (61)2 (21)6 (hyp). Corollary Of the 56 topological types of smooth plane sextics, 42 types are non-dividing, six are dividing, and eight can be dividing or non-dividing. This accounts for all 64 rigid isotopy types in P 27 R \. 6 / 24

7 Polynomials Proposition Each of the 64 rigid isotopy types is realized by a sextic in Z[x, y, z] 6 whose coefficients have abs. value nd x 6 + y 6 + z 6 1 nd x 6 + y 6 z 6 (11) nd 6(x 4 + y 4 z 4 )(x 2 + y 2 2z 2 ) + x 5 y 2 nd (x 4 + y 4 z 4 )((x + 4z) 2 + (y + 4z) 2 z 2 ) + z 6 (21) nd 16((x+z) 2 +(y+z) 2 z 2 )(x 2 +y 2 7z 2 )((x z) 2 +(y z) 2 z 2 ) + x 3 y 3 (11)1 nd ((x + 2z) 2 + (y + 2z) 2 z 2 )(x 2 +y 2 3z 2 )(x 2 +y 2 z 2 ) + x 5 y 3 nd (x 2 + y 2 z 2 )(x 2 + y 2 2z 2 )(x 2 + y 2 3z 2 ) + x 6 (hyp) d 6(x 2 +y 2 z 2 )(x 2 +y 2 2z 2 )(x 2 +y 2 3z 2 )+x 3 y 3 (31) nd (10(x 4 x 3 z+2x 2 y 2 +3xy 2 z+y 4 )+z 4 )(x 2 +y 2 z 2 )+x 5 y (21)1 nd (10(x 4 x 3 z+2x 2 y 2 +3xy 2 z+y 4 )+z 4 )((x+z ) 2 +y 2 2z 2 )+x 5 y (11)2 nd (10(x 4 x 3 z+2x 2 y 2 +3xy 2 z+y 4 )+z 4 )(x 2 +(y z) 2 z 2 )+x 5 y and many more representatives 7 / 24

8 Robinson Sextic Consider this net of sextics: a(x 6 +y 6 +z 6 ) + bx 2 y 2 z 2 + c(x 4 y 2 +x 4 z 2 +x 2 y 4 +x 2 z 4 +y 4 z 2 +y 2 z 4 ). For (a : b : c) = (1 : 3 : 1) this a nonnegative sextic that is not SOS. The discriminant of this net is the following curve of degree 75 in P 2 R : = a 3 (a + c) 6 (3a c) 18 (3a + b + 6c) 4 (3a + b 3c) 8 (9a 3 3a 2 b + ab 2 3ac 2 bc 2 + 2c 3 ) (a : b : c) = (19 : 60 : 20) gives our sextic for the ten ovals type 10d. 8 / 24

9 Eleven Ovals Hilbert (1891) argued that type (51)5 does not exist. Gudkov (1969) showed that Hilbert had made a mistake. (91)1 d ( (yz x 2 )(60(x+z)z (6x+6z y) 2 )+ 118(10x+8y+3z)(12x+32y+z)(12x 32y z)(10x 8y 3z))(x 2 yz) y 6 (51)5 d x x 5 y x 4 y x 3 y x 2 y xy y x 5 z x 4 yz x 3 y 2 z x 2 y 3 z x 2 z x 4 z x 3 yz x 2 y 2 z x 3 z y 4 z x 2 yz xy 4 z y 3 z y 5 z xyz y 2 z yz 5 z 6 (11)9 d (340291(yz x 2 )((x + 2z)z 2(y 2z) 2 ) +(10x 8y 3z)(12x 27y z)(12x + 28y + z)(10x + 7y + 3z))(x 2 yz) + y 6 9 / 24

10 SexticClassifier We wrote fast Mathematica code, using built-in method for cylindrical algebraic decomposition. Its input is a sextic f Z[x, y, z] 6. Its output is the topological type of V R (f ). We computed various empirical distributions. Here is one experiment with 1, 500, 000 samples: (11) 4 (11)1 (21) 5 (11)2 (21)1 6 (31) Table: Topological types sampled from the U(3)-invariant distribution For the uniform distribution on { 10 12,..., } we obtained (11) % 18.24% 2.09% 1.44% 0.65% 0.06% Conclusion: Most types never occur when sampling at random!! 10 / 24

11 Transitions Theorem (Itenberg 1994) For each edge in our poset, both combinatorial transitions (shrinking or fusing) can be realized by a singular curve with exactly one ordinary node. 11 / 24

12 Transitions Theorem For curves of even degree, every discriminantal transition between rigid isotopy types is one of the following: shrinking an ovals, fusing two ovals, and turning an oval inside out. Figure: Type (21)2d transitions into Type (21)2nd by turning inside out. 12 / 24

13 Sylvester s Formula We evaluate as the determinant of a matrix...found in [Gelfand-Kapranov-Zelevinsky]. Each entry in the first 30 columns is either 0 or one of the c ijk. The last 15 columns contain cubics in the c ijk. The matrix is ( R[x, y, z] 3 ) 3 R[x, y, z] 4 R[x, y, z] 8 On the first summand, it maps a triple of cubics to an octic: (a, b, c) a f x + b f y + c f z. On the second summand, it maps a quartic monomial x r y s z t to the octic det(m rst ), where M rst is any 3 3-matrix of ternary forms satisfying f / x f / y = M rst x r+1 y s+1. f / z z t+1 Proposition The discriminant is the determinant of this matrix. 13 / 24

14 Avoidance Locus For a real curve C of even degree d, the avoidance locus A C is the set of lines in P 2 R that do not intersect C R. This is a semialgebraic subset of the dual projective plane (P 2 ) R. Points on C are lines in P 2 tangent to C. The dual curve C has degree d(d 1) in (P 2 ). It divides (P 2 ) R into connected components. Proposition The avoidance locus A C is a union of connected components of (P 2 ) R \C R. Each component appearing in A C is convex. 14 / 24

15 Avoiding Sextics Figure: A smooth sextic of type 10nd whose avoidance locus is empty Proposition Let deg(c) = d even. The number of convex components of the avoidance locus A C is bounded above by d d d d + 1. Corollary For all m {0, 1,..., 46} there exists a smooth sextic C in P 2 R whose avoidance locus A C has exactly m convex components. 15 / 24

16 Avoiding Sextics Figure: A sextic C of type 8nd; its 68 relevant bitangents represent A C. 16 / 24

17 Bitangents and Flexes A general sextic in P 2 C has 324 bitangents and 72 inflection points. Conjecture The number of real bitangents of a smooth sextic in P 2 R ranges from 12 to 306. The lower bound is attained by curves of types 0, 1, (11) and (hyp). The upper bound is attained by (51)5. Transitions: (411) C has an undulation point. (222) C has a tritangent line. (321) C has a flex-bitangent. Theorem The loci (222) and (321) are irreducible hypersurfaces in P 27 of degree 1224 and 306 respectively. Their union is the Zariski closure of the set of smooth sextics having fewer than 324 bitangent lines. 17 / 24

18 Experiments Type Flex Eigenvec Bitang Rank (11) (21) (11) (hyp) (31) (21) (11) (41)nd (41)d (31) (21)2nd (21)2d (11) (51) (41) (31) (21) (11) (61) (51)1nd (51)1d (41) (31)3nd (31)3d (21) Type Flex Eigenvec Bitang Rank (11)5nd (11)5d (71) (61) (51) (41) (31) (21) (11) (81)nd (81)d (71) (61) (51) (41)4nd (41)4d (31) (21) (11) nd d (91) (81) (51) (41) (11) (91) (51) (11) / 24

19 Critical Points on the Sphere A sextic f can have as many as 20 local maxima on the unit sphere S2. The picture shows one with 62 = 2 31 critical points. Its Morse complex is the icosahedron, with f-vector (12, 30, 20). The critical points are the eigenvectors of f. 19 / 24

20 Eigenvectors of Tensors Ternary sextics are symmetric tensors of format Such a symmetric tensor f has 28 distinct entries c ijk. A vector v C 3 is an eigenvector of f if v is parallel to the gradient at v: ( ) x y z rank = 1. f x f y A general f R[x, y, z] d has d 2 d + 1 eigenvectors in P 2 C. Among them are pairs of critical points of f on the sphere S 2. The number of real eigenvectors is 2ω + 1, where ω is the number of ovals (Maccioni 2017). If f is a product of d linear forms, then all complex eigenvectors are real (Abo-Seigal-St 2017). f z Conclusion: The number of real eigenvectors of a general sextic is an odd integer between 3 and 31. Our table summarizes empirical distribution over the 64 types. 20 / 24

21 Ranks of Tensors The rank of a symmetric tensor f R[x, y, z] d is the minimum number of summands in a representation r f (x, y, z) = λ i (a i x + b i y + c i z) d. i=1 For a generic sextic f, the complex rank is 10, and the real rank is between 10 and 19 (Michalek-Moon-St-Ventura 2017). Computing real ranks exactly is very difficult. We used the numerical software tensorlab to determine (our best guess for) the real ranks of the 64 representatives. Type Flex Eigenvec Bitang Rank (11) (21) (11) (41)nd (41)d (31) (21)2nd (21)2d (11) Type Flex Eigenvec Bitang Rank (11)5nd (11)5d (71) (61) (51) (41) (31) (21) (11) (81)nd (81)d / 24 (71)

22 Quartic Surfaces Our 64 sextics represent K3 surfaces over Q. The two basic models for algebraic K3 surfaces are quartic surfaces in P 3 and double-covers of P 2 branched at a sextic curve. A real K3 surface is orientable and has 10 connected components. Its Euler characteristic is between 18 and 20. (Silhol 1989) Can construct quartic surfaces with desired topology from our curves: Example Let F be the quartic 100w w 2 x x w 2 y x 2 y y w 2 yz 75x 2 yz 1552y 3 z w 2 z 2 487x 2 z y 2 z yz z 4. The surface V R (F ) is connected of genus 10, so χ = 20. Example (Rohn 1913) Let G = τ(s1 2 6s 2) 2 + (s1 2 4s 2) 2 64s 4, where s i is the ith elementary symmetric polynomial in x, y, z, w and τ = Then V R (G) consists of 10 spheres, so χ = / 24

23 Conclusion The geometry and topology of real algebraic varieties is a beautiful subject, with many great results, especially from the Russian school. We seek to connect this to current problems and developments in Applied Algebraic Geometry. This requires computational and experimental work with polynomials. We studied explicit sextics like ( (yz x 2 )(60(x+z)z (6x+6z y )2 )+ 118(10x+8y +3z)(12x+32y +z)(12x 32y z)(10x 8y 3z))(x 2 yz) y 6 Page 1 of 1 Quiz: What does the real picture look like for this curve? 23 / 24

24 Eleven Ovals (91)1 d ( (yz x 2 )(60(x+z)z (6x+6z y) 2 )+ 118(10x+8y+3z)(12x+32y+z)(12x 32y z)(10x 8y 3z))(x 2 yz) y 6 (51)5 d x x 5 y x 4 y x 3 y x 2 y xy y x 5 z x 4 yz x 3 y 2 z x 2 y 3 z x 2 z x 4 z x 3 yz x 2 y 2 z x 3 z y 4 z x 2 yz xy 4 z y 3 z y 5 z xyz y 2 z yz 5 z 6 (11)9 d (340291(yz x 2 )((x + 2z)z 2(y 2z) 2 ) +(10x 8y 3z)(12x 27y z)(12x + 28y + z)(10x + 7y + 3z))(x 2 yz) + y 6 24 / 24